Shock interactions with density inhomogeneities: Models and scaling laws for circulation deposition and instability growth rates

Vorticity is deposited baroclinically by shock waves on density inhomogeneities. The main parameters governing circulation deposition are the Mach number of the shock, the density ratio across the interface ($h$), the ratio of specific heats of the two gases and the geometry of the interface. For a slow/fast (s/f) planar interface, circulation deposition can be derived analytically from polar analysis. We derive an exact expression for $s$, circulation deposition per unit unshocked interface length within the regular refraction regime. This is an extension to s/f interfaces of the work done by Samtaney and Zabusky for fast/slow (f/s) interfaces.; The model developed for planar interfaces is then extended to more general interfaces viz. s/f sinusoidally perturbed interfaces, reshocked f/s interfaces and, along with the model for f/s interfaces, to shock interactions with heavy prolate elliptical cylinders. In the last case, we identify two different modes of interactions (leading to different circulation deposition mechanisms) caused by competition between the incident and transmitted shocks on the leeward side of the ellipse.; We also present a comparison of two vorticity-based reduced models for the Richtmyer-Meshkov instability (shock accelerated density-stratified sinusoidal interface). We compare the growth rate of an elliptical patch model (the interface is fitted with a patch) and a sheet model (the interface is approximated with a sheet) versus numerical simulations. We find that the sheet model gives a closer approximation to the numerical results due to its geometric similarity to the actual interface.; Lastly, we investigate the crushing of cylindrical inhomogeneities (of circular and elliptical cross-sections). We identify the generation and evolution of vortex projectiles, which play an important role in the redistribution of the inhomogeneities and can be considered a prototypical mechanism for mixing.; Numerical investigations were conducted by solving the Euler equations with a second-order Godunov method. All models were validated by comparing them with numerical solutions. We find that the models for circulation deposition agree with the simulations within the regular refraction regime while those for the growth-rate of the Richtmyer-Meshkov instability are valid for weak shock interactions.